Re: Well Ordering

Thank you, Thank you, HallsofIvy.

If the concept of well ordering is meaningless for a single digit, and the positive integers are well ordered, then there is no such thing as a single integer subset, so if a subset of the integers contains n, it has to contain a larger number → Archimedes Postulate <-> Euclids Postulate

And also Archimedes postulate (Euclid) implies well ordering.

Now wasn’t that worth 16 posts?

Last edited by Hartlw; January 10th 2013 at 11:38 AM.
Reason: revised conclusion

Re: Well Ordering

Originally Posted by Hartlw

Thank you, Thank you, HallsofIvy.

If the concept of well ordering is meaningless for a single digit, and the positive integers are well ordered, then there is no such thing as a single integer subset, so if a subset of the integers contains n, it has to contain a larger number → Archimedes Postulate <-> Euclids Postulate

And also Archimedes postulate (Euclid) implies well ordering.

Now wasn’t that worth 16 posts?

oh, bother!

the natural numbers come with a CANONICAL well-ordering.

if one uses the construction, s(x) = x U {x}, then k < n iff k is an element of n.

for example:

2 = {0,1}, so 1 < 2, and 0 < 2.

the well-ordering of the natural numbers is "axiomatic", that is: it is INTRINSIC, and equivalent in strength to the axiom schema of induction. actually, ZF set theory says something a bit MORE: there exists an infinite well-ordered set (which may, or may not be, the natural numbers). this stops "just short" of the axiom of choice, in that it does not assert that EVERY set is well-ordered (but certainly implies a method of well-ordering any FINITE set, using an injection into the well-ordered infinite set).

the notion that singleton subsets do not exist is absurd, and violates the axiom of extensionality.

Re: Well Ordering

It's hard to believe this isn't spam. Your original question was "What is the least member of {2}?" That is a set with one member. Whether you ask for "least member", "largest member" with respect to whatever order relation, it must be a member so there is only one possible answer!

Re: Well Ordering

But the concept of order is NOT meaningless for the set of integers. And even a single integer is a member of the set of integers. The smallest member of {2} is 2. In fact it is also the largest member of {2}, the "evenest" member of {2}, and the "*****est" member of {2} as long as "*****" is an attribute that 2 has- because 2 is the only member of the set so any answerable question about a member of {2} must be "2".

The well ordering principle does apply to sets of integer that have a "lower bound". And we don't need to talk about a "glb". If a set of integers has a lower bound, then it has a smallest member, the least member of the set. We usually apply the term "glb" to sets that do NOT have a smallest member. For example, the set of all positive rational numbers does not have a smallest member. It has glb 0 which is not in the set.